Answer :
Let's solve the problem step-by-step by expanding the expression [tex]\((x^2 - 5x)(2x^2 + x - 3)\)[/tex].
### Step 1: Distribute Each Term of the First Polynomial
The expression is a multiplication of two polynomials, so we'll use the distributive property to multiply each term in the first polynomial, [tex]\(x^2 - 5x\)[/tex], by each term in the second polynomial, [tex]\(2x^2 + x - 3\)[/tex].
### Step 2: Multiply [tex]\(x^2\)[/tex] Across the Second Polynomial
- Multiply [tex]\(x^2\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
x^2 \times 2x^2 = 2x^4
\][/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x^2 \times x = x^3
\][/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
x^2 \times (-3) = -3x^2
\][/tex]
### Step 3: Multiply [tex]\(-5x\)[/tex] Across the Second Polynomial
- Multiply [tex]\(-5x\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
-5x \times 2x^2 = -10x^3
\][/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
-5x \times x = -5x^2
\][/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-5x \times (-3) = 15x
\][/tex]
### Step 4: Combine All the Products
Now, we combine all these terms:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
### Step 5: Combine Like Terms
1. Combine [tex]\(x^3\)[/tex] terms:
[tex]\[
x^3 - 10x^3 = -9x^3
\][/tex]
2. Combine [tex]\(x^2\)[/tex] terms:
[tex]\[
-3x^2 - 5x^2 = -8x^2
\][/tex]
Now, we bring the terms together:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
So, the expression [tex]\((x^2 - 5x)(2x^2 + x - 3)\)[/tex] simplifies to [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
Therefore, the correct answer is B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
### Step 1: Distribute Each Term of the First Polynomial
The expression is a multiplication of two polynomials, so we'll use the distributive property to multiply each term in the first polynomial, [tex]\(x^2 - 5x\)[/tex], by each term in the second polynomial, [tex]\(2x^2 + x - 3\)[/tex].
### Step 2: Multiply [tex]\(x^2\)[/tex] Across the Second Polynomial
- Multiply [tex]\(x^2\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
x^2 \times 2x^2 = 2x^4
\][/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x^2 \times x = x^3
\][/tex]
- Multiply [tex]\(x^2\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
x^2 \times (-3) = -3x^2
\][/tex]
### Step 3: Multiply [tex]\(-5x\)[/tex] Across the Second Polynomial
- Multiply [tex]\(-5x\)[/tex] by [tex]\(2x^2\)[/tex]:
[tex]\[
-5x \times 2x^2 = -10x^3
\][/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
-5x \times x = -5x^2
\][/tex]
- Multiply [tex]\(-5x\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-5x \times (-3) = 15x
\][/tex]
### Step 4: Combine All the Products
Now, we combine all these terms:
[tex]\[
2x^4 + x^3 - 3x^2 - 10x^3 - 5x^2 + 15x
\][/tex]
### Step 5: Combine Like Terms
1. Combine [tex]\(x^3\)[/tex] terms:
[tex]\[
x^3 - 10x^3 = -9x^3
\][/tex]
2. Combine [tex]\(x^2\)[/tex] terms:
[tex]\[
-3x^2 - 5x^2 = -8x^2
\][/tex]
Now, we bring the terms together:
[tex]\[
2x^4 - 9x^3 - 8x^2 + 15x
\][/tex]
So, the expression [tex]\((x^2 - 5x)(2x^2 + x - 3)\)[/tex] simplifies to [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
Therefore, the correct answer is B. [tex]\(2x^4 - 9x^3 - 8x^2 + 15x\)[/tex].
